Average Error: 12.2 → 0.4
Time: 25.7s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right), \left(\sqrt{\frac{0.125}{1 - v} \cdot \mathsf{fma}\left(v, -2, 3\right)} \cdot \sqrt{\frac{0.125}{1 - v} \cdot \mathsf{fma}\left(v, -2, 3\right)}\right), 4.5\right)\]
\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5
↓
\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right), \left(\sqrt{\frac{0.125}{1 - v} \cdot \mathsf{fma}\left(v, -2, 3\right)} \cdot \sqrt{\frac{0.125}{1 - v} \cdot \mathsf{fma}\left(v, -2, 3\right)}\right), 4.5\right)
double f(double v, double w, double r) {
        double r766370 = 3.0;
        double r766371 = 2.0;
        double r766372 = r;
        double r766373 = r766372 * r766372;
        double r766374 = r766371 / r766373;
        double r766375 = r766370 + r766374;
        double r766376 = 0.125;
        double r766377 = v;
        double r766378 = r766371 * r766377;
        double r766379 = r766370 - r766378;
        double r766380 = r766376 * r766379;
        double r766381 = w;
        double r766382 = r766381 * r766381;
        double r766383 = r766382 * r766372;
        double r766384 = r766383 * r766372;
        double r766385 = r766380 * r766384;
        double r766386 = 1.0;
        double r766387 = r766386 - r766377;
        double r766388 = r766385 / r766387;
        double r766389 = r766375 - r766388;
        double r766390 = 4.5;
        double r766391 = r766389 - r766390;
        return r766391;
}
↓
double f(double v, double w, double r) {
        double r766392 = 3.0;
        double r766393 = 2.0;
        double r766394 = r;
        double r766395 = r766394 * r766394;
        double r766396 = r766393 / r766395;
        double r766397 = r766392 + r766396;
        double r766398 = w;
        double r766399 = r766398 * r766394;
        double r766400 = r766399 * r766399;
        double r766401 = 0.125;
        double r766402 = 1.0;
        double r766403 = v;
        double r766404 = r766402 - r766403;
        double r766405 = r766401 / r766404;
        double r766406 = -2.0;
        double r766407 = fma(r766403, r766406, r766392);
        double r766408 = r766405 * r766407;
        double r766409 = sqrt(r766408);
        double r766410 = r766409 * r766409;
        double r766411 = 4.5;
        double r766412 = fma(r766400, r766410, r766411);
        double r766413 = r766397 - r766412;
        return r766413;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 12.2

    \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]

  2. Simplified0.4

    \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right), \left(\frac{0.125}{1 - v} \cdot \mathsf{fma}\left(v, -2, 3\right)\right), 4.5\right)}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.4

    \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right), \color{blue}{\left(\sqrt{\frac{0.125}{1 - v} \cdot \mathsf{fma}\left(v, -2, 3\right)} \cdot \sqrt{\frac{0.125}{1 - v} \cdot \mathsf{fma}\left(v, -2, 3\right)}\right)}, 4.5\right)\]

  5. Final simplification0.4

    \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right), \left(\sqrt{\frac{0.125}{1 - v} \cdot \mathsf{fma}\left(v, -2, 3\right)} \cdot \sqrt{\frac{0.125}{1 - v} \cdot \mathsf{fma}\left(v, -2, 3\right)}\right), 4.5\right)\]

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  (- (- (+ 3 (/ 2 (* r r))) (/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))) 4.5))